Ingeniería Biomédica
2025-09-29
A system is a rule that maps an input signal to an output signal. In continuous time and discrete time we depict and denote: \[ x(t)\ \longrightarrow\ y(t),\qquad x[n]\ \longrightarrow\ y[n]. \] These are the standard input–output representations used throughout the text.
We often write the system as an operator ( {} ): \[ y(t)=\mathcal{T}\{x(t)\},\qquad y[n]=\mathcal{T}\{x[n]\}. \] Block diagrams are used to represent systems and interconnections (series/cascade and parallel).
Causal: output depends only on present/past input values. For LTI systems, causality is characterized by the impulse response: \[ \text{Discrete time: } h[n]=0\ \text{for } n<0;\qquad \text{Continuous time: } h(t)=0\ \text{for } t<0. \] Under these conditions, the convolution reduces to depend only on past/present input.
Bounded-Input Bounded-Output (BIBO) stability: bounded input implies bounded output. For LTI systems: \[ \sum_{k=-\infty}^{\infty} |h[k]| < \infty \quad \Longleftrightarrow \quad \text{discrete-time LTI is stable}, \] \[ \int_{-\infty}^{\infty} |h(t)|\,dt < \infty \quad \Longleftrightarrow \quad \text{continuous-time LTI is stable}. \] These are necessary and sufficient conditions.
A system is linear if it satisfies additivity and homogeneity: for any signals (x_1,x_2) and scalar (c), \[ \mathcal{T}\{x_1+x_2\}=\mathcal{T}\{x_1\}+\mathcal{T}\{x_2\},\qquad \mathcal{T}\{c\,x\}=c\,\mathcal{T}\{x\}. \] (These two conditions together are equivalent to linearity.)
A system is time-invariant if a shift in the input produces an identical shift in the output: \[ \mathcal{T}\{x(t-t_0)\}=y(t-t_0),\ \text{whenever}\ y(t)=\mathcal{T}\{x(t)\}. \] Analogously in discrete time with shifts by integer indices. (Definition used throughout the text in system properties and LTI analysis.)
A system is invertible if distinct inputs produce distinct outputs; equivalently, there exists an inverse system that recovers the input from the output. Example pair (discrete time): the accumulator and the first-difference operator are inverses: \[ y[n]=\sum_{k=-\infty}^{n} x[k] \quad \Longleftrightarrow \quad x[n]=y[n]-y[n-1]. \]
For LTI systems, interconnections admit simple algebraic descriptions via transforms: e.g., in the Laplace domain, series and parallel lead to product and sum of system functions, respectively: \[ H_{\text{series}}(s)=H_1(s)H_2(s),\qquad H_{\text{parallel}}(s)=H_1(s)+H_2(s). \]
Importante
The digital filter separates the noise and the information of a discrete signal.
Suppose a discrete time system \[ y[n] = \sum_{k=1}^{K} a_k y[n - k] + \sum_{m=0}^{M} b_m x[n - m]\]
K y M are the order of the filter.
We must know the initial condition.
Gain
\[y[n] = G x[n]\]
Delay of \(n_0\) samples
\[y[n] = x[n - n_0]\]
Two points moving average
\[y[n] = \frac{1}{2} (x[n] + x[n - 1])\]
Euler approximation of the derivative
\[y[n] = \frac{x[n] - x[n - 1]}{T_s}\]
Averaging over N consecutive epochs of duration L
\[y[n] = \frac{1}{N} \sum_{k=0}^{N-1} x[n - kL]\]
Trapezoidal integration formula
\[y[n] = y[n - 1] + \frac{T_s}{2} (x[n] + x[n - 1])\]
Digital “leaky integrator” (First-order lowpass filter)
\[y[n] = a y[n - 1] + x[n], \quad 0 < a < 1\]
Digital resonator (Second-order system)
\[y[n] = a_1 y[n - 1] + a_2 y[n - 2] + b x[n], \quad a_1^2 + 4a_2 < 0\]
For a system’s response to be fully described by its impulse response, the system must satisfy the following key conditions.
Linearity
If the system responds to \(x_1[n]\) with \(y_1[n]\) and to \(x_2[n]\) with \(y_2[n]\), then:
\[y[n] = y_1[n] + y_2[n]\]
Homogeneity
If the input is scaled by a constant \(c\), the output is also scaled:
\[\text{If } x[n] \rightarrow y[n], \text{ then } cx[n] \rightarrow cy[n]\]
Time Invariance
A system must be time-invariant, meaning a time shift in the input causes the same shift in the output:
\[\text{If } x[n] \rightarrow y[n], \text{ then } x[n - n_0] \rightarrow y[n - n_0]\]
Causality
A causal system is one where the output at time \(n\) depends only on present and past inputs:
\[h[n] = 0 \quad \forall n < 0\]
Stability
If the impulse response does not satisfy this condition, the system may produce unbounded outputs.
\[\sum_{n=-\infty}^{\infty} |h[n]| < \infty\]
Convolution Representation
If all condition met then \[y[n] = x[n] * h[n] = \sum_{m=-\infty}^{\infty} x[m] h[n - m]\]